Some quadrature formulae with nonstandard weights - Core

Journal of Computational and Applied Mathematics 235 (2010) 602–614

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Some quadrature formulae with nonstandard weights G. Mastroianni ∗ , D. Occorsio Dipartimento di Matematica ed Informatica, Università degli Studi della Basilicata, Via dell’Ateneo Lucano 10, 85100 Potenza, Italy

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abstract

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Article history: Received 15 April 2008 Received in revised form 11 December 2009 MSC: 65D30 41A05 41A10

In this paper the authors study ‘‘truncated’’ quadrature rules based on the zeros of Generalized Laguerre polynomials. Then, they prove the stability and the convergence of the introduced integration rules. Some numerical tests confirm the theoretical results. © 2010 Elsevier B.V. All rights reserved.

Keywords: Quadrature formulae Orthogonal polynomials Approximation by polynomials

1. Introduction This paper is devoted to the computation of integrals of the kind ∞

Z

∞

Z

f (x)w(x)dx, 0

f (x)K (x, y)u(x)dx 0

where f is a continuous function in (0, +∞) having a possible exponential growth for x → +∞, K (x, y) is a kernel defined β

β

in R+ and u(x) = xα e−x /2 (1 + x)λ , w(x) = xα e−x , x ∈ R+ , α > −1, β > 12 , γ > −1, λ ≥ 0. The weights w and u are not classical ones, however we shall continue to call them Generalized Laguerre weights. We propose the computation of the first integral by using a Gaussian-type quadrature rule (see [1]) and we will estimate the error when the function f is continuous or belongs to some Sobolev space. To evaluate the second integral we introduce the product rule (8), based on a special Lagrange projector. We give necessary and sufficient conditions under which such kinds of formulae are stable and convergent. These are the main results of the paper and they are given in Theorem 3.2 and Corollary 3.3. In order to show the performance of our method we will propose some numerical examples, comparing the proposed truncated rules with other methods. The plan of the paper is as follows: the next section contains some preliminary results and notations; Section 3 includes the main results, while Section 4 contains some numerical tests. Finally in Section 5 the proofs of the main results are supplied. 2

2. Preliminaries 2.1. Functional spaces β

The space Lp := Lp (R+ ), 1 ≤ p ≤ +∞ is definedR in the usual way and, with u(x) = xγ e−x /2 (1 + x)λ , λ ≥ 0, γ > − , β > 12 , we will say f ∈ Lpu iff fu ∈ Lp , i.e. kfukpp = R+ |(fu)(x)|p dx < +∞. When p = ∞ we define for γ ≥ 0, 1 p

∗

Corresponding author. E-mail addresses: [email protected] (G. Mastroianni), [email protected] (D. Occorsio).

0377-0427/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2010.06.011

G. Mastroianni, D. Occorsio / Journal of Computational and Applied Mathematics 235 (2010) 602–614

C u = L∞ u

603

= f ∈ C 0 (R+ ), lim f (x)u(x) = 0 = lim f (x)u(x) x→0+

x→+∞

being C 0 (R+ ) the space of the continuous functions in R+ . The norm in Cu is kf kCu = supx≥0 |(fu)(x)| = kfuk∞ . The Sobolev space of order r ≥ 1 is defined as

Wrp (u) = f ∈ Lpu : f (r −1) ∈ AC (R+ ), kf (r ) ϕ r ukp < +∞ ,

ϕ(x) =

√

x,

1 ≤ p ≤ +∞

where AC (R ) is the set of all absolutely continuous functions in R , equipped with the norm +

+

kf kWrp (u) = kfukp + kf (r ) ϕ r ukp . In what follows the following modulus of continuity will be useful [2]

Ωϕr (f , t )u,p = sup ku∆rhϕ f kLp (Irh ) ,

Irh = [8r 2 h2 , Ah∗ ],

0 −1.

2.2. Lagrange interpolation β

Let w(x) = xα e−x , α > −1, β > positive leading coefficients, i.e. pm (x) = γm xm + · · · ,

1 , 2

and let {pm }m be the corresponding orthonormal polynomials sequence with

γm > 0 and

+∞

Z

pm (x)pn (x)w(x)dx = δm,n ,

m = 0, 1, . . . .

0

For any function f denote by L¯ m+1 (f ) the Lagrange polynomial interpolating f at the zeros x1 , . . . , xm of pm and on the extra knot bm which is the M–R–S number w.r.t. the weight w . It is m +1

L¯ m+1 (f , x) =

X

¯lk (x)f (xk ),

(3)

k =1

where pm (x) ¯lk (x) = lk (x) bm − x , k = 1, . . . , m, lk (x) = 0 , bm − xk pm (xk )(x − xk ) ¯lm+1 (x) = pm (x) . pm (bm ) We remark that L¯ m+1 (f , x) projects Cu into Pm . Following an idea in [4], for any fixed 0 < θ < 1, and defined xj = min {xk ≥ θ bm } ,

(4)

1≤k≤m

we define the set

Pm∗ = q ∈ Pm : q(xi ) = q(bm ) = 0, xi > xj ⊂ Pm .

604


Setting fj = f χj , where χj is the characteristic function of the interval [0, xj ], we define L∗m+1 (f , x) := L¯ m+1 (fj , x) =

j X

¯lk (x)f (xk ).

(5)

k=1

L∗m+1 (f , x) interpolates f at the knots x1 , . . . , xj . Moreover, for k > j, it results L∗m+1 (f , xk ) = L∗m+1 (f , bm ) = 0. Therefore S ∗ L∗m+1 (f , x) ∈ Pm∗ and it projects Cu into Pm∗ . A basis for Pm∗ is {¯l1 , ¯l2 , . . . , ¯lj }. Now we prove that m Pm is dense in p Lu , 1 ≤ p ≤ +∞. Indeed, setting E˜ m (f )u,p := inf k(f − Q )ukp , ∗ Q ∈Pm

the next lemma estimates E˜ m (f )u,p in terms of EM (f )u,p , where M ∼ m is a proper fraction of m.

h

Lemma 2.1. Let PM ∈ PM be a polynomial of best approximation of f ∈ Cu , with M = m parameters γ , λ, α, β >

1 2

θ β 1+θ

i

∼ m. For any choice of the

and 1 ≤ p ≤ +∞, we have

E˜ m (f )u,p ≤ k[f − L∗m+1 (w, PM )]ukp ≤ C EM (f )u,p + e−Am kfukp ,

(6)

and

√

am

√

r

m

(r ) r

k[Lm+1 (w, PM )] ϕ ukp ≤ C ∗

am m

Z 0

where ϕ(x) =

Ωϕr (f , t )u,p t

dt + C e−Am kfukp

√

x, C 6= C (m, f ).

3. Main results Now we consider two quadrature rules based on the previous interpolation process. The first is a Gaussian-type formula, +∞

Z

f (x)w(x)dx = 0

j X

G f (xk )λk + em (f ) =: Im (f ) + em (f ),

(7)

k=1

m where {xk }m k=1 are the zeros of pm , {λk (w)}k=1 are the Christoffel numbers and em (f ) is the remainder term. The second is the so called ‘‘product rule’’

+∞

Z

f (x)K (x, y)u(x)dx = 0

j X

f (xk )Ak (y) + e∗m (f , y) =: Im (f , y) + e∗m (f , y)

(8)

k =1

R +∞

where K : [0, +∞) × [0, +∞) → R, Ak (y) = 0 ¯lk (x)K (x, y)u(x)dx, and e∗m (f ) is the error of the quadrature rule. Truncated quadrature formulae for Laguerre and Freud weights appeared for the first time in [5,1,6–8]. We observe that both of the proposed quadrature rules require only j = j(m) evaluations of the function f at the interpolation nodes and, as a consequence, the possible overflow, when the function f has an exponential growth, is avoided. We remark also that, except some special cases (for instance β = 1), the coefficients in the three term recurrence relation for {pm }m are not explicitly known. On the other hand efficient numerical procedures exist in order to compute the zeros of the orthogonal polynomials j and Christoffel numbers (see for instance [9,10]). The main effort is required in order to compute the coefficients {Ak }k=1 in (8), since they depend on the kernel K . Some details about their construction in the case β = 1 will be given in the last section of the paper. In this section we will discuss the convergence and the stability of the previous rules. About the formula (7), recently it was proved in [11] that, if f ∈ Cσ , σ (x) = (1 + x)λ xγ e−ax , 0 < a ≤ 1, and, moreover +∞

Z 0

w(x) dx < +∞, σ (x )

(9)

then

|em (f )| ≤ C EM (f )σ ,∞ + e−AM kf σ k∞ , h i θ β where M = 1−θ m ∼ m, C 6= C (m, f ) and A 6= A(m, f ). Notice that EM (f )u,∞ can be estimated by (2). In the next theorem we estimate the error of the Gaussian rule in terms of Sobolev norm of the function f :

(10)


605

Theorem 3.1. Let r ≥ 1. For any f ∈ Wr1 (w) we have

|em (f )| ≤ C

√

bm

r

m

kf kWr1 (w) ,

(11)

where C 6= C (m, f ). We remark that, by an argument in [6], it is easy to prove that (11) does not hold if in (7) j is replaced by m, i.e. using the ordinary Gaussian rule. Next theorem deals with the stability and the convergence of the rule (8). Setting log+ f (x) = log (max(1, |f (x)|)) , we state Theorem 3.2. Assume that the function K : [0, +∞) × [0, +∞) → R and the weights u and w satisfy the conditions

√ w(x)ϕ(x) ≤ C < +∞, C 6= C (m, f ) u(x) x≥0 Z +∞ u(x) |K (x, y)| 1 + log+ x + log+ |K (x, y)| dx < +∞. sup √ w(x)ϕ(x) y≥0 0 sup

(12)

(13)

Then, for any function f s.t. kfuk∞ < +∞, it results sup |Im (f , y)| ≤ C kfuk∞ ,

C 6= C (m, f ).

(14)

u(x) |K (x, y)|dx < +∞. √ w(x)ϕ(x)

(15)

y≥0

Moreover, (14) implies +∞

Z sup y≥0

0

Finally, for any function f ∈ Cu ,

|e∗m (f )| ≤ C EM (f )u,∞ + e−Am k fu k∞ , (16) h i θ β where M = 1−θ m ∼ m and the positive constants C and A are independent of m, f . In particular, if f ∈ Wr∞ (u), it results |em (f )| ≤ C ∗

√

bm

m

r

k f kWr∞ (u) ,

(17)

1

where bm ∼ m β and C 6= C (m, f ). Remark. It is useful to observe that (14) implies the stability of the product rule, i.e. sup

j X |Ak (y)|

y≥0 k=1

u( x k )

< +∞.

We remark also that if the kernel K (x, y) is a constant, by virtue of (15), Theorem 3.2 does not hold. On the other hand, for some special kernels of the kind K˜ (x, y) =

Q (x, y) =

P ( x, y ) Q (x, y) n Y

,

P (x, y) =

|x − yk |ζk ±

j =1

s Y

|x − tk |γk ±

j =1 l Y

l Y

|y − zk |δk

(18)

j=1

|y − vk |µk > 0,

j=1

where tk , zk , yk , vk , γk , δk , ζk , µk ∈ R+ , Theorem 3.2 specializes into the following corollary. Corollary 3.3. If K˜ (x, y) is defined in (18) and (12) holds, then, for any f s.t kfuk∞ < +∞, it results sup |Im (f , y)| ≤ C kfuk∞ m

(19)

606


Table 1 7 7 2 2 Example 1. g (x) = arctan(x) 2 e−x /2+x , f (x) = arctan(x) 2 ex /2 .

j

G Im (f )

m

Gm (g )

22 43 84 168 –

0.566 0.5660242 0.56602429838788 0.566024298387888 –

32 64 128 256 512

0.56602 0.56602429 0.5660242983 0.56602429838 0.56602429838

if and only if +∞

Z sup y≥0

0

u(x) |K˜ (x, y)|dx < +∞. √ w(x)ϕ(x)

Moreover (16) and (17) hold true. 4. Numerical examples Now we propose some test using truncated quadrature rules (7) and (8). All the computations are performed in double machine precision 2.2204 × 10−16 , except which concerns the zeros and the Christoffel numbers w.r.t. the weight w with β 6= 1. Indeed, in this case they were computed by the package ‘‘Orthogonal Polynomials’’ (see [9]) in MATHEMATICA, which works by using ‘‘high’’ variable precision. In all the proposed examples each table contains the values of the integrals obtained with j knots instead of m knots, where j = j(m) is the integer defined in (4). We will compare our results with those obtained by using the ordinary Gaussian rule on Laguerre zeros, i.e. m X

+∞

Z

g (x)ρ(x)dx = Gm (g ) + em (g ) = 0

λk (ρ)g (zk ) + em (g ),

k=1

where ρ(x) = e−x xρ , {zk }m k=1 are the zeros of pm (ρ). 4.1. Gaussian rule Example 1. +∞

Z

7

arctan(x) 2

I1 =

√ xe

−x2 2

dx

(20)

0

w(x) =

√

2

xe−x ,

7

f (x) = arctan(x) 2 e

x2 2

∈ W91 (w). C (r ,f )

. In this case we compare our results with those obtained by the √ 7 x2 ordinary Gaussian rule Gm (g ) w.r.t. the weight ρ(x) = e−x x and the function g (x) = arctan(x) 2 e− 2 +x (see Table 1). As According to the estimate (11), the error behaves like

m9/2

we can see, by using our procedure the machine precision is attained with 168 function computations, while the ordinary Gaussian rule is standing in eleven digits for increasing values of m. Example 2. +∞

Z

sin(5x)e−

I2 =

x3 2

√ xdx

(21)

0

w(x) =

√

3

xe−x ,

f (x) = sin(5x)e

x3 2

. 1

Here f is very smooth, since f ∈ Wr1 (w), ∀r ≥ 1. Since bm ∼ m 3 , the error behaves like out that the seminorm rapidly increases with r (for instance, with r = 10, kf denotes the ordinary Gauss–Laguerre rule w.r.t. the weight ρ(x) = e Then we obtain Table 2.

√ −x

(r ) r

C (r ,f ) 5r

, with r ‘‘large’’. We only point

m6

ϕ wk1 ∼ 3.9 × 106 ). In this case Gm (g ) 3

x x and g (x) = sin(5x)e− 2 +x .

4.2. Product rule About the computation of the coefficients in the product rule, we give some details in the Appendix.


607

Table 2 3 3 Example 2. g (x) = sin(5x)e−x /2+x , f (x) = sin(5x)ex /2 .

j

G Im (f )

m

Gm (g )

14 27 43 104 187 –

3.46e−2 3.4698e−2 3.4698839168e−2 3.46988391684420e−2 3.469883916844207e−2 –

16 32 64 128 256 512

2.7e−1 3.1e−2 3.49e−2 3.4697e−2 3.4698839e−2 3.469883916844e−2

Table 3 Example 3. g (x, y) = |x − y|0.1 cos(x), f (x) = cos(x). y = 0.1

y=1

y = 10

j

Im (f , 0.1)

m

Gm (g , 0.1)

j

Im (f , 1)

m

Gm (g , 1)

j

Im (f , 10)

m

Gm (g , 10)

13 25 49 97 192

0.4040 0.404083 0.404083318312 0.40408331831278 0.404083318312782

16 32 64 128 256

0.38 0.41 0.402 0.407 0.406

13 25 49 97 192

0.452 0.4527789 0.45277897088 0.452778970891112 0.4527789708911126

16 32 64 128 256

0.45 0.452 0.455 0.454 0.453

13 25 49 97 192

0.629815 0.62981519 0.6298151957983 0.629815195798344 0.629815195798344

16 32 64 128 256

0.62 0.6298 0.62981 0.6298 0.62981

Table 4 Example 4. g (x, y) =

sin(x) x+y

, f (x) = sin(x).

y = 0.1

y = 0.01

y = 0.001

j

Im (f , 0.1)

m

Gm (g , 0.1)

j

Im (f , 0.01)

m

Gm (g , 0.01) j

Im (f , 0.001)

m

Gm (g , 0.001)

11 20 40 79 – –

0.5959 0.5959258 0.59592584070747 0.595925840707471 – –

16 32 64 128 256 512

0.59 0.59 0.5959 0.59592 0.59592584 0.595925840707

11 20 40 79 – –

0.74 0.74591 0.7459180963011 0.74591809630114 – –

16 32 64 128 256 512

0.7 0.7 0.74 0.746 0.746 0.7459

0.7 0.7791 0.7791921087034 0.77919210870343 – –

16 32 64 128 256 512

0.7 0.78 0.78 0.78 0.779 0.779

11 20 40 79 – –

Example 3. +∞

Z

cos x|x − y|0.1 e−x dx

(22)

0

f (x) = cos(x),

K (x, y) = e−x/2 |x − y|0.1 ,

α = −0.5, γ = 0, λ = 0.

All the assumptions of the Theorem 3.2 are satisfied. In this example we compare the results obtained by the proposed truncated product rule with those obtained by using the Gaussian rule Gm (g , y) w.r.t. the weight ρ(x) = e−x and the function C (r ,f ) g (x, y) = cos x|x−y|0.1 . According to the estimate (17), since f ∈ Wr∞ (u), ∀r ≥ 1, the error behaves like r , with r ‘‘large’’. m2

Table 3 contains the approximate values of the integral for three different choices of y. Example 4. +∞

Z 0

sin x −x e dx

(23)

x+y

f (x) = sin(x),

K ( x, y ) =

e− x / 2 x+y

,

α = −0.5, γ = 0, λ = 0.

All the assumptions of the Theorem 3.2 are satisfied. Here Gm (g , y) represents the Gaussian rule w.r.t. the weight C (r ,f ) sin x . According to the estimate (17), since f ∈ Wr∞ (u), ∀r ≥ 1, the error behaves like r , x+y

ρ(x) = e−x and g (x, y) =

m2

with r ‘‘large’’. Table 4 contains the approximate values of the integral for three different choices of y. In this example, even though the kernel is smooth, the Gaussian rule gives a poor approximation for ‘‘small’’ values of y. 5. Proofs of the main results 5.1. Polynomial inequalities Let Pm ∈ Pm . With 1 ≤ p < +∞, in [2] it was proved a Remez-type inequality

kPm ukLp (R+ ) ≤ C kPm ukLp ( am ≤x≤am ) , m2

(24)

608


and

kPm ukLp (x≥am (1+δ)) ≤ C1 e−C2 m kPm ukLp (0,am ) ,

1 ≤ p ≤ +∞,

(25)

where δ > 0 is fixed and the constants C , C1 , C2 are independent of m and Pm . We need the Bernstein inequality

√

am

r

m

kq(mr ) ϕ r ukp ≤ C kqm ukp ,

qm ∈ Pm , 0 < p ≤ +∞,

C 6= C (m)

(26)

and the following Nikolskii inequality [2]

kqm uk∞ ≤ C

m

2p

kqm ukp ,

am

qm ∈ Pm , 1 ≤ p < +∞,

C 6= C (m).

(27)

5.2. Orthogonal polynomials Now we collect some properties and estimates for the polynomials pm . These estimates can be deduced from the analogous ones in [12] with a change of variable (see also [13]). For the Christoffel numbers λk , k = 1, . . . , m, w.r.t. the weight w , the following estimate holds β

λk ∼ xαk e−xk 1xk ,

k = 1, 2, . . . , m,

1xk = xk+1 − xk .

(28)

Moreover

√ 1

p 4 ∼ ∆ xk bm xk , √ |p0m (xk )| w(xk )

bm √ xk ,

1xk ∼

m

k ≤ j.

(29)

Let x ∈ [x1 , xm ] and d = d(x) ∈ {1, . . . , m} be the index of a zero of pm (wα ) closest to x. Then, for some positive constant C 6= C (m, x, d), we have 1

C

x − xd xd − xd±1

2 ≤

p2m

(x)e

−x β

x+

bm

α+ 12 q

− 31

|bm − x| + bm m

m2

≤C

x − xd xd − xd±1

2 (30)

and for a fixed real number 0 < δ < 1,

p |pm (x)| w(x) ≤ C

1

√ 4

q

− 31

4

x |bm − x| + bm m

,

bm m2

≤ x ≤ bm (1 + δ).

(31)

In particular

p 1 |pm (x)| w(x) ≤ C √ 4

bm x

,

bm m2

≤ x ≤ θ bm , 0 < θ < 1 .

(32)

Proof of Lemma 2.1. First we prove (6)

k[f − L∗m+1 (w, PM )]ukp ≤ k(f − PM )ukp + k[PM − L∗m+1 (PM )]ukp ≤ EM (f )u,p + k[Lm+1 (PM ) − L∗m+1 (PM )]ukp .

(33)

By (5) u( x ) , |¯lk (x)| u (xk ) k=j+1 m+1

|L¯ m+1 (PM , x) − L∗m+1 (PM , x)|u(x) ≤ kPM ukL∞ (θ am ,+∞) max x≥0

X

and, taking into account (29) and (31), we get

3 γ − α2 − 14 m +1 X u(x) bm − x 4 1xk x 1+x ¯ max |lk (x)| ≤ ≤ C mτ , x≥0 u ( x ) b − x x 1 + x | x − x | k m k k k k k=1 k=1 m+1

X

(34)

for some τ > 0 and the other parameters γ , α arbitrarily fixed. By using (25)

|L¯ m+1 (PM , x) − L∗m+1 (PM , x)|u(x) ≤ C mτ e−Am kPM ukL∞ (0,am (1+δ)) and by (27)

(35)


τ + 2p

m

k[L¯ m+1 (PM ) − L∗m+1 (PM )]ukp ≤ C e−Am

1 p

kPM ukp .

609

(36)

am Therefore

k[f − L∗m+1 (PM )]ukp ≤ EM (f )u,p + C e−Am kfukp .

(37)

To prove (7), let us start from

√

am

r

m

k[L∗m+1 (PM )](r ) ϕ r ukp ≤

√

am

r n

m

o k[L∗m+1 (PM ) − L¯ m+1 (PM )](r ) ϕ r ukp + kPM(r ) ϕ r ukp .

(38)

Using the estimate in [2]

√

am

r

(r ) r

√

am

kPM ϕ ukp ≤ C ωϕ f ,

m

r

m

u ,p

,

and by (26), we have:

√

am

r

m

(

√

k[L∗m+1 (PM )](r ) ϕ r ukp ≤ C k[L∗m+1 (PM ) − L¯ m+1 (w, PM )]ukp + ωϕr f ,

Combining (36) with the last inequality, (7) follows.

am

m

)

w,p

.

(39)

Proof of Theorem 3.1. First we prove

|rm (f )| ≤ C

√

bm

kf 0 ϕwk1 + e−Bm kf wk1

m

√

≤C

bm m

kf 0 ϕwk1 .

(40)

Let Q = L∗m+1 (w 2 , PM ) where PM ∈ PM is a polynomial such that k[f − PM ]wk1 ≤ C EM (f )w,1 . We have rm (f ) = rm (f − Q ) and by (7),

|rm (f − Q )| ≤

+∞

Z

|f (x) − Q (x)|w(x)dx +

0

j X

λk (w)|f (xk ) − Q (xk )|.

(41)

k=1

Using (28) and w(x) ∼ w(y), |x − y| ≤ C 1xk , we get

|f (xk )|1xk ≤

xk+1

Z

√ bm

|f (t )|dt +

m

xk

xk+1

Z

√ |f 0 (t )| tdt

xk

and j X

λk (w)|f (xk )| ≤ C

√

xj+1

Z

|f (t )|w(t )dt +

x1

k=1

Therefore

bm m

xj+1

Z

|f (t )0 |ϕ(t )w(t )dt .

x1

√

|rm (f − P )| ≤ C EM (f )w,1 + C

bm m

kL∗m+1 (w 2 , PM )0 ϕwk1 + kf 0 ϕwk1 .

Using then (7) and [2]

ωϕ f , 1

√ bm m

√

bm

≤C

m

w,1

kf wk1 + kf 0 ϕwk1 ,

we have

√

( |rm (f − P )| ≤ C EM (f )w,1 +

bm m

kf ϕwk1 + e 0

−Am

kf wk1 + ωϕ f , 1

√ bm m

)

w,1

√ ≤C

bm m

kf wk1 + kf 0 ϕwk1 .

Assume now r > 1. Let Q ∈ PM , such that k[f − Q ]wk1 = EM (f )w,1 . We have rm (f ) = rm (f − Q ) + rm (Q ).

(42)

610


By (42)

√ bm

|rm (f − Q )| ≤ C

k(f − Q )wk1 + k(f − Q )0 ϕwk1 .

m

(43)

Since Q is a polynomial of best approximation in L1w , f −Q =

+∞ X (Q2k+1 M − Q2k M ) k=0

holds almost everywhere in (0, +∞). So we have, by virtue of (26)

√

√ bm

m

+∞ bm X

k(f − Q )0 ϕwk1 ≤ C ≤C

m

k=0

+∞ X

2k+1

p k=0

≤ 2C

k(Q2k+1 M − Q2k M )0 ϕwk1

b2k+1

k(Q2k+1 M − Q2k M )wk1

+∞ X 2k+1 p E2k+1 M (f )w,1

+∞ X 2k+1 p ≤ 2C

b2k+1

k=0

√

bM

≤C

r kf

M

√

bM

≤C

p

b2k+1 M

bM

r

M

!r kf (r ) ϕ r wk1

2k+1 M +∞ X

(r ) r

ϕ wk1

p

b2k+1

r

M

! r −1 (46)

2k+1

k=0

kf (r ) ϕ r wk1 ,

where in the last inequality we use r ≥ 2 and β >

|rm (f − Q )| ≤ C

(45)

b2k+1

k=0

√

(44)

1 . 2

(47)

Therefore

(r ) r kf ϕ wk1 + kf wk1 .

(48)

Consider now rm (Q )

Z j j m X +∞ X X Qm (xk ) − Qm (xk ) Qm (xk )λk = Qm (x)w(x)dx − |rm (Q )| = 0 k=1 k=1 k=1 X λ (w) X k λk Q (xk ) ≤ = max |Q (x)|w(x) ≤ C max |Q (x)|w(x). k>j w(xk ) x≥θ bm k>j x≥θ bm Using (28)

|rmG (Q )| ≤ C bm e−Am kQ wk∞ and using inequality (27)

| (Q )| ≤ C e G rm

−Am

2

m bm

bm kQ wk1 ≤ C e−Am kf wk1 .

The theorem follows combining last estimate with (48).

In order to prove Theorem 3.2, we need the following Lemma 5.1. Let g a measurable function and A(x) = 1

Z

A(x)|g (x)|dx ≥ C 0

√ 4

√

x(bm − x)|pm (x)| w(x). We have

1

Z

|g (x)|dx 0

where C 6= C (m, g ). Proof. Let δ¯ > 0 be ‘‘small’’. Define δk = 3δ 1xk = 3δ (xk+1 − xk ), and

(49)


611

Im = ∪x1 ≤xk ≤1 ([xk − δk , xk + δk ]). To prove (49), set CIm = [0, 1] \ Im . By (30) we get

p p x − xd 4 ≥ C, |pm (w, x)| w(x) x(bm − x) ≥ C xd − xd±1

x ∈ CIm ,

and consequently 1

Z

A(x)|g (x)|dx ≥ C

Z

0

|g (x)|dx. CIm

Since the measure of Im is bounded by δ¯ , for a suitable δ¯ , we conclude 1

Z

A(x)|g (x)|dx ≥ C

1

Z

|g (x)|dx. 0

0

Proof of Theorem 3.2. First we prove that (12) and (13) imply (14), which is true if

kL∗m+1 (f )K (·, y)uk ≤ C kfuk∞ ,

C 6= C (m, f )

(50)

holds. In the proof we will replace bm with am since am ∼ bm . Start from

∗

L

m+1

(f )K (·, y)u 1 = L∗m+1 (f )K (·, y)u L (0,am ) + L∗m+1 (f )K (·, y)u L (am ,am (1+δ)) 1 1

∗

+ Lm+1 (wα , f )K (·, y)u L (am (1+δ),+∞) =: I1 + I2 + I3 , 1

(51)

where δ is a fixed real number, 0 < δ < 1. Setting gm = sgn(L∗m+1 (f )), am

Z

L∗m+1 (f , t )K (t , y)gm (t )u(t )dt

I1 = 0

∼

j X

f (xk )

k=1

p0m (xk )(am − xk )

am

Z 0

j X (am − x)pm (x) f (xk ) K (x, y)gm (x)u(x)dx = Π (xk ), 0 (x − xk ) p (x )(am − xk ) k=1 m k

where

Π (t ) =

am

Z 0

[(am − x)pm (x)q(x) − (am − t )pm (t )q(t )] K (x, y)gm (x)u(x) dx (x − t ) q(x)

and q is an arbitrary polynomial of degree lm, l fixed. Using (29) we obtain

kfuk |f (xk )| ≤ C 3/4 |p0m (xk )(am − xk )| am

√ w(xk )ϕ(xk ) ∆ xk u(xk )

and I1 ≤

C 3/4

kfuk∞

bm

j X k=1

√ w(xk )ϕ(xk ) 1xk |Π (xk )|. u( x k )

Since Π is a polynomial of degree m + ml, by a Marcinkiewicz-type inequality in [14] we have

√ Z am √ w(t )ϕ(t ) w(t )ϕ(t ) I1 ≤ 3/4 kfuk∞ |Π (t )|dt ≤ C kfuk∞ u(t ) u( t ) am x1 x1 Z am Z am Fm (x) K (x, y)gm (x)u(x) × dx + Gm (t ) dx dt =: C kfuk∞ {Σ1 + Σ2 } x−t (x − t )q(x) 0 0 Z

C

am

(52)

with Fm (x) = Gm (x) = Using

p

1−

1−

x

43 q

am x am

34

p u(x) w(x) x(am − x)pm (x) √ K (x, y)gm (x), w(x)ϕ(x)

pm (x)q(x)

√ 4

am − x .

√ w(x) x(am − x)|pm (x)| ≤ C for x ≤ am , we have

u(x) |Fm (x)| ≤ C √ |K (x, y)|, w(x)ϕ(x)

|G(x)| ≤ C √

u(x)

w(x)ϕ(x)

.

(53)

612


√

Since under the assumptions (12) the function

w(t )ϕ(t ) u(t )

is bounded in [0, +∞) and using the boundedness of the Hilbert

transform for any function h : kh log hkL1 (a,b) < +∞ [15] +

b

Z a

Z

h(ξ )

b

z−ξ

a

dξ dz ≤ C + C

b

Z

h(z )[1 + log+ |h(z )| + log+ z ]dz ,

(54)

a

we get am

Z

Σ1 ≤ C

|Fm (t )| 1 + log+ |Fm (t )| + log+ t dt

(55)

u(t ) |K (t , y)| 1 + log+ |K (t , y)| + log+ t dt < +∞, √ w(t )ϕ(t )

(56)

0

and using (53)

Σ1 ≤ C

am

Z 0

where the last bound follows taking into account (13). β Choosing q(x) ∼ e−x /2 xγ (see [2]) it follows

Σ2 ≤

Z

√ Z Z am Z +∞ w(t )ϕ(t ) +∞ K (x, y)gm (x)u(x) K (x, y)gm (x)u(x) dt . dx dt ≤ C dx u( t ) q(x)(x − t ) q(x)(x − t ) 0 x1 0

am x1

In view of (54),

Σ2 ≤

Z

am

|K (x, y)| 1 + log+ |K (x, y)| + log+ x dx < +∞

(57)

x1

taking into account the assumptions (12)–(13). Combining (55) and (57) with (52) it follows I1 ≤ C kfuk∞ . By (29) am (1+δ)

Z I2 =

|L∗m+1 (f , t )u(t )K (t , y)|dt

am

≤C

j kfuk∞ X 3

am4

k=1

√ Z am (1+δ) (am − x)|pm (x)K (x, y)u(x)| w(xk )ϕ(xk ) 1 xk dx. u(xk ) (x − xk ) am

Taking into account the assumption (12), being (x − xk ) ≥ am and am (1+δ) q

Z I2 ≤ C kfuk∞

Pj

k=1

1xk ≤ am , we have

p α 1 w(x) x|am − x||pm (x)K (x, y)|xγ − 2 − 4 dx

am

+∞

Z ≤ C kfuk∞ 0

u( x ) |K (x, y)| < +∞ √ w(x)ϕ(x)

where last bound follows taking into account the assumption (13). Consider now I3 . I3 = L∗m+1 (f )K (·, y)u L (a (1+δ),+∞) ≤ L∗m+1 (f )u L (a (1+δ),+∞) kK (·, y)k1 . ∞ m 1 m

Using (25) and taking into account the assumption (12)–(13), I3 ≤ C e−C2 m L∗m+1 (f )u L (0,a ) , ∞ m

and by (34), we have I3 ≤ C e−C2 m mτ kfuk∞ .

(58)

Therefore (14) is completely proved. Now we prove (14) implies (15). Consider the function f0 (x) s. t. f0 (xk ) = sgn(p0m (xk )(x − xk )), x1 ≤ xk ≤ 1 and f0 (xk ) = 0, xk > 1, |f0 (x)| ≤ 1. Therefore we have L∗m+1 (f0 , x)K (x, y)u(x) =

X

am − x

pm (x)u(x)

a − xk |p0m (xk )(x − xk )| x1 ≤xk ≤1 m

3 √ 4 X am − x 4 w(xk )√ p p xk ∆xk α 1 ≥ C 4 x(am − x)|pm (x)| w(x)xγ − 2 − 4 K (x, y) . am − x k |x − xk | x ≤x ≤1 1

k


Since for 0 ≤ x ≤ 1, |x − xk | ≤ 1, and using (am − x)/(am − xk ) ∼ 1, setting A(x) = L∗m+1 (f0 , x)K (x, y)u(x) ≥ C A(x) √

u(x)

w(x)ϕ(x)

X

|K (x, y)

613

√ 4

√ x(am − x)|pm (x)| w(x), we obtain

p √ 1xk w(xk ) 4 xk

x1 ≤xk ≤1

u(x)|K (x, y)|

≥ C A(x) √ , w(x)ϕ(x) since

X

√ 4

p

1xk w(xk ) xk ≥

x1 ≤xk ≤1

1

Z 1 2

√ w(x) 4 xdx.

Therefore, by (14) we have +∞

Z

|Lm+1 (f0 , x)K (x, y)u(x)|dx ≥ C ∗

k uk ∞ ≥ C

1

Z 0

0

u(x)K (x, y) A(x) √ dx w(x)ϕ(x)

and by Lemma 5.1 1

Z k uk ∞ ≥ C

α

1

xγ − 2 − 4 |K (x, y)|dx =

1

Z

0

0

u(x) dx. |K (x, y)| √ w(x)ϕ(x)

Now we prove (16). Let be Pm ∈ Pm∗ . We have

|e∗m (f , y)| ≤ k[f − L∗m+1 (Pm )]K (·, y)uk1 + kL∗m+1 ((f − Pm ))K (·, y)uk1 ≤ k[f − L∗m+1 (Pm )]uk∞ kK (·, y)k1 + kL∗m+1 ((f − Pm ))K (·, y)uk1 where we use (50), and taking into account Lemma 2.1, (16) follows. Finally (17) follows by (1) and (2).

Acknowledgements The authors are grateful to the referees for their useful suggestions and remarks. This research was supported by University of Basilicata (Italy). Appendix In this section we give some details about the construction of the coefficients in the product rule (8). In the special case

λ = 0, β = 1, the coefficients {Ak (y)}jk=1 have the following expression Z +∞ 1 x Ak (y) = lk (x)(bm − x)K (x, y)e− 2 xγ dx bm − xk 0 Z +∞ −1 X λk (w) m x = pj (x) pj (x)(bm − x)K (x, y)e− 2 xγ dx bm − xk j=0 0 =

−1 n o X λk (w) m γ γ +1 pj (x) bm Mj (y) − Mj (y) bm − xk j=0

where γ

Mj (y) =

+∞

Z

x

pj (wα , x)K (x, y)xγ e− 2 ,

0 x

are the so called modified moments w.r.t. the weight xγ e− 2 and the kernel K . Recurrence relations for the next two kernels were deduced in [16] and here we recall them for the convenience of the reader. First of all we recall the three term recurrence relation w.r.t. the weight w(x) = e−x xα . p−1 (wα , x) = 0,

p0 (wα , x) = √

1

Γ (α + 1) an+1 pp n+1 (wα ; x) = (x − en )pn (wα , x) − an pn−1 (wα , x) an =

n(n + α)en = 2n + α + 1.

Consider now x K1 (x, y) = e− 2 (x + y)ρ ,

Mnγ (y, ρ) =

+∞

Z 0

ρ ∈ R, y > 0, x

pn (wα , x)K1 (x, y)xγ e− 2 dx.

(59)

614


We have

 γ M−1 (y, ρ) = 0     γ  M0 (y, ρ) = √

1

y(γ +ρ)/2 ey/2 Γ (γ + 1)W ρ−γ , ρ+γ +1 (y) 2 2 Γ (α + 1) γ γ an+1 Mn+1 (y, ρ) = −(en + t )Mnγ (y, ρ) − an Mn−1 (y, ρ) + Mnγ (y, ρ + 1)    γ γ γ   an+1 Mn+1 (y, ρ + 1) = gn Mn (y, ρ + 1) − y(ρ + 1)Mn (y, ρ) n > 0 g n = 2 + γ + ρ + n − en

(60)

where Wk,µ (y) is the second Whittaker function. x Setting K2 (x, y) = e− 2 |x − y|µ ,

Mn (y, µ) =

+∞

Z

µ > −1 y > 0 and x

pn (wα , x)K2 (x, y)xγ e− 2 dx,

0

Mn− (y, µ) =

t

Z

(t − x)µ pn (wα , x)xγ e−x dx

0

Mn+ (y, µ) =

+∞

Z

(x − t )µ pn (wα , x)xγ e−x dx

t

it is Mn (y, µ) = Mn− (y, µ) + Mn+ (y, µ)

M (y, µ) = 0 −1   1   B(µ + 1, γ + 1)yγ +1+µ 1 F1 (γ + 1; γ + µ + 2; −y) M0− (y, µ) = √    Γ (α + 1)    Γ (µ + 1) −y/2 (ga+mu)/2   e y W γ −µ ,− µ+γ +1 (y) M0+ (y, µ) = √ 2 2 Γ (α + 1) − − − a M ( y , µ) = ( y − e ) M ( y , µ) + M ( y , µ + 1) − an Mn−−1 (y, µ)  n + 1 n n n n + 1   − −  a M ( y , µ + 1 ) = − h M ( y , µ + 1 ) − y (µ + 1 )Mn− (y, µ)  n+1 n+1 n n   + + +  an+1 Mn+1 (y, µ) = (y − en )Mn (y, µ) − Mn (y, µ + 1) − an Mn+−1 (y, µ)    an+1 Mn++1 (y, µ + 1) = hn Mn+ (y, µ + 1) − y(µ + 1)Mn+ (y, µ)  n > 0 hn = n + α + γ − µ − 1 .

(61)

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